PHYS08200604017 Manimala Mitra - Homi Bhabha National Institute

We next discuss the neutrino oscillation parameters, the three angles in the U PMNS
mixing matrix and two mass square differences ∆m 2 21 and ∆m 2 31. Note that with the mass
matrixm ν giveninEq. (4.43), i.etakingonlytheeffect oftripletYukawa contributioninto
account one would get the three following mass eigenvalues for the three light standard
model neutrinos,
and the eigenvectors
⎛
1
√ ⎝ −Y ⎞
Σ 3
0 ⎠,
YΣ 2 1
+YΣ 2 3
Y Σ1
m 1 = 0, m 2 = 0, m 3 = v2 2
2M (Y 2 Σ 1
+Y 2 Σ 2
+Y 2 Σ 3
).
⎛
1
√ ⎝ −Y ⎞
Σ 2
Y Σ1
⎠,
YΣ 2 1
+YΣ 2 2
0
⎛
1
√ ⎝ Y ⎞
Σ 1
Y Σ2
⎠
YΣ 2 1
+YΣ 2 2
Y Σ3
respectively. Clearly, two of the light neutrinos emerge as massless while the third one
gets mass, which is in conflict with the low energy neutrino data. Similarly if one has only
matrix B which comes as a consequence of R-parity violation then one also would obtain
only one light neutrino to be massive, in general determining only the largest atmospheric
mass scale [20,21] . However the simultaneous presence of the matrix A, B and C in Eq.
(4.42) make a second eigenvalue non-zero, while the third one remains zero. With the
choice ũ as of the same order of u, the third term in Eq. (4.42) would be suppressed
compared to the first two terms. Hence the simultaneous presence of the matrix A and
B are very crucial to get both the solar and atmospheric mass splitting and the allowed
oscillation parameters. Eigenvalues of the full M ν given in Eq. (4.42) are
m 1 = 0, m 2,3 ∼ 1 2 [W ∓√ W 2 −V], (4.47)
where
and
W = v2 2
2M
∑
YΣ 2 i
+ µα 2
i
∑
u 2 i + ũv √
1α ∑
u i Y Σi , (4.48)
2
i i
V = 4( v2 2 µα
4M − ũ2 v1 2α2
)[YΣ 2 8 1
(u 2 2 +u2 3 )+Y2 Σ 2
(u 2 3 +u2 1 )+Y2 Σ 3
(u 2 1 +u2 2 ) (4.49)
−2(u 1 u 2 Y Σ1 Y Σ2 +u 1 u 3 Y Σ1 Y Σ3 +u 2 u 3 Y Σ2 Y Σ3 )].
Similarly we have obtained approximate analytic expressions for the mixing matrix, however
the expressions obtained are too complicated and hence we do not present them
here. Instead we show in Tables 4.1 and Table 4.2 an example set of model parameters
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